Multivariate spatial central limit theorems with applications to percolation and spatial graphs

Annals of Probability

Volumn 33, Issue 5, 2005, Pages 1945-1991

  Penrose, Mathew D ^a,b  

^a UNIVERSITY OF BATH   (United Kingdom)

^b University of Bath   (United Kingdom)

Author keywords

Central limit theorem; Empirical process; Minimal spanning tree; On line nearest neighbor graph; Percolation; White noise

Indexed keywords

EID: 27644525072     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/009117905000000206     Document Type: Article

References (27)

1. ALDOUS, D. and STEELE, J. M. (1992). Asymptotics for Euclidean minimal spanning trees on random points. Probab. Theory Related Fields 92 247-258.
2. BARYSHNIKOV, Y. and YUKICH, J. E. (2005). Gaussian limits for random measures in geometric probability. Ann. Appl. Probab. 19 213-253.
3. BERGER, N., BOLLOBÁS, B., BORGS, C., CHAYES, J. and RIORDAN, O. (2003). Degree distribution of the FKP model. Automata, Languages and Programming: 30th International Colloquium, ICALP 2003. Lecture Notes in Comput. Sci. 2719 725-738. Springer, New York.
4. COX, J. T. and GRIMMETT, G. (1984). Central limit theorems for associated random variables and the percolation model. Ann. Probab. 12 514-528.
5. DURRETT, R. (1991). Probability: Theory and Examples. Wadsworth & Brooks/Cole, Pacific Grove, CA.
6. FABRIKANT, A., KOUTSOUPIAS, E. and PAPADIMITRIOU, C. M. (2002). Heuristically optimized trade-offs: A new paradigm for power laws in the Internet. ICALP 2002. Lecture Notes in Comput. Sci. 2380 110-122. Springer, New York.
7. GRIMMETT, G. (1999). Percolation, 2nd ed. Springer, Berlin.
8. HEINRICH, L. and MOLCHANOV, I. S. (1999). Central limit theorem for a class of random measures associated with germ-grain models. Adv. in Appl. Probab. 31 283-314.
9. KESTEN, H. and LEE, S. (1996). The central limit theorem for weighted minimal spanning trees on random points. Ann. Appl. Probab. 6 495-527.
10. LEE, S. (1997). The central limit theorem for Euclidean minimal spanning trees I. Ann. Appl. Probab. 7 996-1020.
11. LEE, S. (1999). The central limit theorem for Euclidean minimal spanning trees II. Adv. in Appl. Probab. 31 969-984.
12. MCLEISH, D. L. (1974). Dependent central limit theorems and invariance principles. Ann. Probab. 2 620-628.
13. MEESTER, R. and ROY, R. (1995). Continuum Percolation. Cambridge Univ. Press.
14. PENROSE, M. D. (2001). A central limit theorem with applications to percolation, epidemics and Boolean models. Ann. Probab. 29 1515-1546.
15. PENROSE, M. (2003). Random Geometric Graphs. Oxford Univ. Press, New York.
16. PENROSE, M. D. and PISZTORA, A. (1996). Large deviations for discrete and continuous percolation. Adv. in Appl. Probab. 28 29-52.
17. PENROSE, M. D. and WADE, A. R. (2004). On the total length of the random minimal directed spanning tree. Preprint. Available at http://arxiv.org.
18. PENROSE, M. D. and YUKICH, J. E. (2001). Central limit theorems for some graphs in computational geometry. Ann. Appl. Probab. 11 1005-1041.
19. PENROSE, M. D. and YUKICH, J. E. (2003). Weak laws of large numbers in geometric probability. Ann. Appl. Probab. 13 277-303.
20. PENROSE, M. D. and YUKICH, J. E. (2004). Normal approximation in geometric probability. In Stein's Method and Applications (A. D. Barbour and L. H. Y. Chen, eds.) 37-58. World Scientific, Singapore.
21. ROY, R. and SARKAR, A. (2003). High density asymptotics of the Poisson random connection model. Phys. A 318 230-242.
22. RUDIN, W. (1976). Principles of Mathematical Analysis, 3rd ed. McGraw-Hill, New York.
23. RUDIN, W. (1987). Real and Complex Analysis, 3rd ed. McGraw-Hill, New York.
24. WILLIAMS, D. (1991). Probability with Martingales. Cambridge Univ. Press.
25. YUKICH, J. E. (1998). Probability Theory of Classical Euclidean Optimization Problems. Lecture Notes in Math. 1675. Springer, New York.
26. YUKICH, J. E. (1999). Asymptotics for weighted minimal spanning trees on random points. Stochastic Process. Appl. 85 123-138.
27. d lattice. J. Theoret. Probab. 14 165-187.